# Summation of a series $a_1 + a_1 a_2 + a_1 a_2 a_3 + a_1 a_2 a_3 a_4 + \cdots$

If $$a_1 > 0$$ and $$a_{n+1} = \ln\left[\frac{\exp(a_n)-1}{a_n}\right]$$ then what is the value of $$a_1 + a_1 a_2 + a_1 a_2 a_3 + a_1 a_2 a_3 a_4 + \cdots?$$

I have proved that this sequence is a decreasing one and converging to zero, but can't figure out what to do next. Any leads appreciated.

• Please use MathJax to type your mathematical expressions; here is a tutorial:math.meta.stackexchange.com/questions/5020/… As written it is difficult to understand your question. Dec 2, 2019 at 1:08
• What exactly do you mean by "value of ..."? If the infinite series converges, then it has a finite real value. Is that sufficient to get what you want to prove? The value clearly is a function of $a_1$. Do you want an answer in "closed form"? Dec 2, 2019 at 2:58
• @Somos Will it blow your mind if I tell you the answer is $e^{a_1} - 1$? Dec 2, 2019 at 3:14
• If that is true, then please mention that in your question. Dec 2, 2019 at 3:17
• @Somos That's not my question, rather my answer. Dec 2, 2019 at 3:18

Writing $$b_n = e^{a_n} - 1$$, we have $$b_{n + 1} + 1 = b_n / a_n$$, or $$b_n = a_n + a_n b_{n + 1}$$.

By induction, we have $$b_1 = a_1 + a_1a_2 + \dotsc + a_1a_2\dotsc a_n + a_1a_2\dotsc a_nb_{n + 1}$$ for any $$n \geq 1$$.

Letting $$n$$ tends to infinity, and noting that $$\lim\limits_{n\rightarrow \infty} a_n = \lim\limits_{n\rightarrow \infty}b_n = 0$$, we see that the infinite sum converges to $$b_1 = e^{a_1} - 1$$.